File size: 2,676 Bytes
9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Probability.ClassicalON.DyadicSine
namespace OAI
noncomputable section
open scoped BigOperators ComplexConjugate
namespace ClassicalON
variable {k : ℕ}
def groupedPoint (m s : Bool) (p : DyadicFreq k) : ℤ × ℤ :=
let q := if m then (dyadicPoint p).swap else dyadicPoint p
if s then -q else q
def groupedTorusPoint (m s : Bool) (p : DyadicFreq k) : DiscreteTorus (16*2^k) :=
(((groupedPoint m s p).1:ZMod (16*2^k)), ((groupedPoint m s p).2:ZMod (16*2^k)))
instance dyadicModulus_neZero (k : ℕ) : NeZero (16*2^k) := ⟨by positivity⟩
theorem groupedTorusPoint_injective (m s : Bool) :
Function.Injective (groupedTorusPoint (k := k) m s) := by
have hs := dyadicTorusPoint_injective (k := k)
cases m <;> cases s <;> intro p q h <;> apply hs
· exact h
· simpa [groupedTorusPoint, groupedPoint, dyadicTorusPoint, Prod.ext_iff] using h
· exact Prod.ext (congrArg Prod.snd h) (congrArg Prod.fst h)
· have h' : (dyadicTorusPoint p).swap = (dyadicTorusPoint q).swap := by
simpa [groupedTorusPoint, groupedPoint, dyadicTorusPoint, Prod.ext_iff] using h
exact Prod.swap_injective h'
def dyadicWeight (p : DyadicFreq k) : ℝ :=
1/((dyadicRadius p:ℝ)*Real.sqrt (dyadicRadius p:ℝ))
theorem dyadicWeight_pos (p : DyadicFreq k) : 0 < dyadicWeight p := by
have h : (0:ℝ)<dyadicRadius p := by exact_mod_cast dyadicRadius_pos p
unfold dyadicWeight
positivity
@[simp] theorem dyadicWeight_sq (p : DyadicFreq k) :
dyadicWeight p^2 = (1/(dyadicRadius p:ℝ))^3 := by
have hr : (0:ℝ)<dyadicRadius p := by exact_mod_cast dyadicRadius_pos p
unfold dyadicWeight
rw [div_pow, mul_pow, Real.sq_sqrt hr.le]
field_simp
@[simp] theorem dyadicWeight_mul_sqrt (p : DyadicFreq k) :
dyadicWeight p*Real.sqrt (dyadicRadius p:ℝ) = 1/(dyadicRadius p:ℝ) := by
have hr : (0:ℝ)<dyadicRadius p := by exact_mod_cast dyadicRadius_pos p
have hs : Real.sqrt (dyadicRadius p:ℝ) ≠ 0 := (Real.sqrt_pos.mpr hr).ne'
unfold dyadicWeight
field_simp
theorem wave_commutator (a b c θ : ℝ) (w : ℂ) (hw : ‖w‖=1) :
((a*b:ℝ):ℂ)*w * conj (((a*c:ℝ):ℂ)*w*Complex.exp (Complex.I*(θ:ℂ))) / 2 -
conj (((a*b:ℝ):ℂ)*w) * (((a*c:ℝ):ℂ)*w*Complex.exp (Complex.I*(θ:ℂ))) / 2 =
-Complex.I * ((a^2*b*c*Real.sin θ:ℝ):ℂ) := by
have hww : w*conj w=1 := by rw [Complex.mul_conj, Complex.normSq_eq_norm_sq, hw]; norm_num
rw [mul_comm Complex.I (θ:ℂ), Complex.exp_ofReal_mul_I]
simp only [map_mul, map_add, Complex.conj_ofReal, Complex.conj_I]
push_cast
linear_combination -Complex.I*(a:ℂ)^2*(b:ℂ)*(c:ℂ)*(Complex.sin (θ:ℂ)) * hww
end ClassicalON
end
end OAI
|